In the past it has been necessary to use a single large conversion matrix to convert from a first frame of reference of a gyro assembly to another frame of reference, for the gyro assembly. Such a prior art method is quite cumbersome and is not amenable to programming on a computer.
The present invention allows one to program a computer in order to convert between non-adjacent frames of reference for a gyro assembly. An acceleration vector with respect to a first frame of reference is converted to an acceleration vector with respect to a second frame of reference. Step wise conversion is continued in order to go from the second frame of reference to a third frame of reference. Such stepwise conversion can be continued to go to a fourth frame of reference, and on to other frames of reference.
The method is based on Euler transformation technique. However, Euler showed only a conversion between two adjacent frames of reference. He did not suggest using the technique to make successive conversions between non-adjacent frames of reference.
Thus in the disclosed method, if one is going from a first frame of reference to a fourth frame of reference, one would use total of nine single axis rotation steps. Three steps would be used in going from the first frame of reference to a second frame of reference. Three steps would be used in going from the second frame of reference to the third frame of reference. Three steps would also be used in going from the third frame of reference to the fourth frame of reference. These nine steps are more easily programmed into a computer than would be three large conversion matrix multiplications, to go from the first frame of reference to the fourth frame of reference.
In the past it was commonly considered necessary to represent the transformation between a given reference frame and another, whose angular rate with respect to the first was known, by a direction cosine matrix, in order to ensure complete freedom of motion of the moving frame. While the direction cosine approach is a valid solution to the transformation problem, there are applications where it becomes burdensome.
One such application involves the use of inertially stabilized autonavigators. In this application, the navigator has its instruments mounted on a stabilized table attached to a vehicle by a four gimbal system whose angles can be read by means of resolvers or other angle readout devices. These gimbals have measurable mismount and non-orthogonally errors which must be compensated to accurately ascertain the transformation between the vehicle frame and the instrument frame. Gyro pickoff angles are nulled to maintain the instrument table reasonably fixed with respect to inertial space but known error in the pickoff angles must be compensated to determine the transformation from the instrument frame to a frame determined by the gyro spin axes. The gyro spin axes also tend to "drift" with respect to inertial space and the transformation from the gyro frame to the inertial frame must be established during alignment and thereafter extrapolated using drift compensation rates. The earth rotates about its polar axis with respect to inertial space and the transformation between these frames is extrapolated with the known spin rate of the earth. Finally, a locally level coordinate frame with a known bearing with respect to North is defined. This frame has an angular rate with respect to the Earth which may be computed from the vehicle velocity with respect to the Earth which, in turn, is computed from inertial accelerations transformed to the locally level frame from the instrument frame where they are sensed by accelerators. A desired output of an autonavigator is usually the attitude of the vehicle expressed in the form of three successive single axis rotations simulating older earth stabilized autonavigators which delivered roll, pitch, and heading directly from the gimbal angle readouts. The cycle of transformations is now complete from vehicle frame to vehicle frame.
Using conventional techniques, the gyro to inertial transformation and the earth to locally level transformation is represented by direction cosine matrices and extrapolated accordingly. The vehicle to instrument table transformation is determined from small angle error compensations and the single axis gimbal angle rotations and converted to a direction cosine matrix. The instrument to gyro transformation is represented by a small angle direction cosine matrix and the inertial to earth transformation by a single axis rotations in the form of a direction cosine matrix. The transformation from the instrument table to the locally level frame is formed by multiplying the direction cosine matrices for instrument to gyro, gyro to inertial, inertial to earth, and earth to locally level. The transformation from vehicle to locally level is formed by multiplying the vehicle to instrument table direction cosine matrix times the instrument table to locally level direction cosine matrix and then extracting the vehicle's roll, pitch, and bearing. Latitude and longitude are extracted from the earth to locally level direction cosines. These processes are cumbersome and consume a great deal of processor throughput.
The method employed by the invention uses four single axis rotations to define both the earth to locally level transformation and the gyro to inertial transformation.
The need for direction cosine matrices is, of course, contradicted by the use of four gimbals on a stabilized instrument table in order to ensure the same end. Leonard Euler (1707-1783) pointed out that any transformation between two coordinate frames can be represented by a series of three single axis rotations. This ignores the rate problems which arise when attempting to build the physical analog of the transformation, the three gimbal structure. The use of the four gimbal structure was the physical solution to the rate problem. The use of four single axis rotations is the solution to representing the transformation between the two coordinate frames discussed above. One of the four angles is designated the "redundant" angle and assumes two states, usually zero and ninety arcdeg. Judiciously "flipping" from one state to the other can resolve the rate problem.
The small angle compensation terms are represented by sequential single axis small angle rotations. The gimbal axis readouts already are single axis rotations. The complete cycle of transformations, therefore, from vehicle to instrument table to gyro to inertial to earth to locally level and finally back to vehicle though roll, pitch, and heading is defined as one continuous sequence of single axis rotations. A single computational entity (or subroutine) may be used to transform any desired vector from any coordinate frame to any other coordinate frame by successive single axis rotations. Furthermore, the latitude and longitude are directly extrapolated except when in polar regions when the desired transverse latitude and longitude replace them. Roll and pitch are computed by transforming the vertical vector (which is a coordinate axis) from the locally level coordinate frame to the vehicle frame in the inverse order where roll and pitch may be extracted. Heading is calculated by transforming the vehicle roll axis (also a coordinate axis) from the vehicle frame to the locally level frame where bearing is extracted and added to the directly extrapolated bearing of the locally level frame to form heading or transverse heading. This solution has been demonstrated in practice to offer large improvements in processor throughput and to also lend itself to clarity in exposition of the autonavigators mechanization equations.
The technique also makes use of certain properties of single axis rotations which allow the rotation to be specified by the names of the axes whose components will undergo change. Furthermore, the order in which the axes are specified allow the rotation to be either positive or negative and interchanging the order of axis specification allows the inverse rotation to be defined. These properties allow a single subroutine to be used to transform vectors from one coordinate frame to another when coupled with a simple table defining the single axis rotations and containing the sine and cosine of each rotation. The sine and cosine of each angle may be calculated at whatever rate is appropriate to that rotation.
The technique also includes simple techniques for the solution of single axis rotation angles which can be derived from known rotation angles.
The Euler Angle Technique consists of the following combinations of facts used to form a single unified technique in the handling of complex coordinate transformation problems.
1. All transformations may be expressed by at most four properly ordered single axis rotations. PA0 2. A single subroutine may be used to transform any vector from one coordinated frame to another. PA0 3. A table defining the rotations and including the sines and cosines of the rotation angles is the core of the technique. PA0 4. The sines and cosines of the rotation angles may be updated independently at whatever rate is appropriate to the accuracy of the solution. PA0 5. Up to three unknown angles in a cycle of rotations may be easily determined if the others are known.